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Plane-Wave Basis

A plane-wave basis represents the Bloch spinors at each \(\boldsymbol k\) point of a periodic one-electron problem in an orthonormal reciprocal-space basis. Its accuracy is controlled primarily by an energy cutoff, and its algebra is independent of atomic positions. Isolated systems enter this framework through a periodic supercell, not through a change of boundary condition. Hartree atomic units are used unless stated otherwise.

Basis and Normalization

Let \(\Omega\) be the volume of the unit cell, \(\mathcal L^*\) its reciprocal lattice, \(\boldsymbol k\) a Bloch wavevector, \(\boldsymbol G\in\mathcal L^*\) a reciprocal-lattice vector, and \(s\in\{\uparrow,\downarrow\}\) a spin index. The plane-wave spinor basis function \(\lvert\boldsymbol G,\boldsymbol k;s\rangle\) is defined by its real-space value

\[ \langle\boldsymbol r \vert\boldsymbol G,\boldsymbol k;s\rangle = \frac{1}{\sqrt{\Omega}} \mathrm e^{ +\mathrm i(\boldsymbol k+\boldsymbol G)\cdot\boldsymbol r } \lvert s\rangle, \]

where the factor \(1/\sqrt{\Omega}\) implements the cell normalization of Periodic Systems. At fixed \(\boldsymbol k\), these basis functions are orthonormal,

\[ \langle \boldsymbol G,\boldsymbol k;s \vert \boldsymbol G',\boldsymbol k;s' \rangle_\Omega = \delta_{\boldsymbol G\boldsymbol G'} \delta_{ss'}. \]

A Bloch spinor, with \(a\) labeling the band or one-electron state, is expanded as

\[ \lvert\psi_{a\boldsymbol k}\rangle = \sum_{\boldsymbol Gs} c_{\boldsymbol Gs,a}(\boldsymbol k) \lvert\boldsymbol G,\boldsymbol k;s\rangle. \]

When the overlap matrix is the identity, cell normalization gives

\[ \sum_{\boldsymbol Gs} c_{\boldsymbol Gs,a}^*(\boldsymbol k) c_{\boldsymbol Gs,b}(\boldsymbol k) = \delta_{ab}. \]

PAW instead normalizes coefficients with the overlap matrix in the generalized eigenproblem introduced below.

The kinetic-energy cutoff \(E_{\mathrm{cut}}\) defines the retained set of reciprocal vectors

\[ \mathcal G_{\boldsymbol k}(E_{\mathrm{cut}}) = \left\{ \boldsymbol G\in\mathcal L^* \;\middle|\; \frac{1}{2} \lvert\boldsymbol k+\boldsymbol G\rvert^2 \le E_{\mathrm{cut}} \right\}. \]

The retained set of \(\boldsymbol G\) vectors can therefore depend weakly on \(\boldsymbol k\). Increasing \(E_{\mathrm{cut}}\) monotonically enlarges the basis for a fixed cell and \(\boldsymbol k\). Its spatial size is

\[ N_{\mathrm{PW}}(\boldsymbol k) = \left| \mathcal G_{\boldsymbol k}(E_{\mathrm{cut}}) \right|. \]

Explicit spin doubles the coefficient-space dimension.

Kohn–Sham Matrix

The plane-wave basis overlap is the identity. In an all-electron or norm-conserving pseudopotential formulation, this also gives an ordinary Hermitian Kohn–Sham eigenproblem. PAW instead introduces a separate overlap matrix \(\mathbf S(\boldsymbol k)\). The common form covering both cases is

\[ \mathbf H(\boldsymbol k) \mathbf C(\boldsymbol k) = \mathbf S(\boldsymbol k) \mathbf C(\boldsymbol k) \mathbf E(\boldsymbol k), \qquad \mathbf C(\boldsymbol k)^\dagger \mathbf S(\boldsymbol k) \mathbf C(\boldsymbol k) = \mathbf I. \]

Here \(\mathbf S(\boldsymbol k)\) is the plane-wave overlap matrix, with elements obtained from \(\hat S\):

\[ S_{\boldsymbol Gs,\boldsymbol G's'}(\boldsymbol k) \equiv \langle \boldsymbol G,\boldsymbol k;s \lvert \hat S \rvert \boldsymbol G',\boldsymbol k;s' \rangle_\Omega. \]

When \(\hat S=\hat I\), \(\mathbf S(\boldsymbol k)=\mathbf I\). The operator represented by \(\mathbf H\) depends on the formulation: it is the corresponding Kohn–Sham Hamiltonian in an all-electron or norm-conserving problem, and the transformed Hamiltonian \(\hat{\widetilde H}=\hat{\mathcal T}^\dagger\hat H_{\mathrm{KS}}\hat{\mathcal T}\) in PAW. These operators are defined in Kohn–Sham Density Functional Theory, Pseudopotentials and Effective-Core Operators, and Projector-Augmented-Wave Method.

The columns of \(\mathbf C(\boldsymbol k)\) are the plane-wave coefficient vectors, and \(\mathbf E(\boldsymbol k)=\operatorname{diag}(\varepsilon_{a\boldsymbol k})\) is the diagonal eigenvalue matrix.

The local part of the effective one-electron potential is the \(2\times2\) spin matrix

\[ \mathbf v_{\mathrm{eff}}(\boldsymbol r) = \mathbf v_{\mathrm{ext}}^{\mathrm{loc}}(\boldsymbol r) +v_{\mathrm H}(\boldsymbol r)\sigma_0 +\mathbf v_{\mathrm{xc}}(\boldsymbol r). \]

All remaining nonlocal terms are collected in the operator \(\hat V_{\mathrm{NL}}\).

The kinetic-energy matrix is diagonal, the local potential couples plane waves through their reciprocal-vector difference, and the nonlocal matrix element is

\[ V_{\boldsymbol Gs,\boldsymbol G's'}^{\mathrm{NL}}(\boldsymbol k) \equiv \langle \boldsymbol G,\boldsymbol k;s \lvert \hat V_{\mathrm{NL}} \rvert \boldsymbol G',\boldsymbol k;s' \rangle_\Omega. \]

The complete Hamiltonian matrix is therefore

\[ H_{\boldsymbol Gs,\boldsymbol G's'}(\boldsymbol k) = \frac{1}{2} \lvert\boldsymbol k+\boldsymbol G\rvert^2 \delta_{\boldsymbol G\boldsymbol G'} \delta_{ss'} + \left[ \mathbf v_{\mathrm{eff}} (\boldsymbol G-\boldsymbol G') \right]_{ss'} + V_{\boldsymbol Gs,\boldsymbol G's'}^{\mathrm{NL}}(\boldsymbol k). \]

For a reciprocal-lattice vector \(\boldsymbol Q\), define the Fourier coefficient of the local potential by

\[ \mathbf v_{\mathrm{eff}}(\boldsymbol Q) = \frac{1}{\Omega} \int_\Omega \mathrm d\boldsymbol r\, \mathrm e^{ -\mathrm i\boldsymbol Q\cdot\boldsymbol r } \mathbf v_{\mathrm{eff}}(\boldsymbol r). \]

The corresponding expansion is

\[ \mathbf v_{\mathrm{eff}}(\boldsymbol r) = \sum_{\boldsymbol Q} \mathrm e^{ +\mathrm i\boldsymbol Q\cdot\boldsymbol r } \mathbf v_{\mathrm{eff}}(\boldsymbol Q). \]

The local contribution is a reciprocal-space convolution and is normally applied through real-space and reciprocal-space transforms. Nonlocal pseudopotential, effective-core spin–orbit, and PAW projector terms are low-rank or projector-structured additions rather than local convolutions.

Density Matrix and Density

The density matrix at each \(\boldsymbol k\) point is

\[ \mathbf D(\boldsymbol k) = \mathbf C(\boldsymbol k) \mathbf f(\boldsymbol k) \mathbf C(\boldsymbol k)^\dagger, \]

where \(\mathbf f(\boldsymbol k)=\operatorname{diag}(f_{a\boldsymbol k})\) is the occupation matrix. In components,

\[ D_{\boldsymbol Gs,\boldsymbol G's'}(\boldsymbol k) = \sum_a f_{a\boldsymbol k} c_{\boldsymbol Gs,a}(\boldsymbol k) c_{\boldsymbol G's',a}^*(\boldsymbol k). \]

The electron number per cell and the expectation value per cell of a one-electron operator with matrix \(\mathbf X(\boldsymbol k)\) are

\[ N_{\mathrm e} = \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \operatorname{tr} \left[ \mathbf D(\boldsymbol k) \mathbf S(\boldsymbol k) \right], \]
\[ \langle\hat X\rangle_{\mathrm{cell}} = \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \operatorname{tr} \left[ \mathbf D(\boldsymbol k) \mathbf X(\boldsymbol k) \right]. \]

Here \(\operatorname{tr}\) runs over the retained reciprocal vectors and the explicit spin indices. In PAW, \(\mathbf X(\boldsymbol k)\) must be the matrix of the corresponding transformed operator, including its one-center correction when present.

When \(\mathbf S(\boldsymbol k)=\mathbf I\), the electron count reduces to \(\Omega_{\mathrm{BZ}}^{-1}\int_{\mathrm{BZ}}\mathrm d\boldsymbol k\, \operatorname{tr}\mathbf D(\boldsymbol k)\).

When the density is represented directly by the plane-wave coefficients, the diagonal real-space kernel gives the local spin-density matrix. Its components are

\[ n_{ss'}(\boldsymbol r) = \frac{1}{\Omega\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \sum_{\boldsymbol G\boldsymbol G'} D_{\boldsymbol Gs,\boldsymbol G's'}(\boldsymbol k) \mathrm e^{ +\mathrm i(\boldsymbol G-\boldsymbol G') \cdot\boldsymbol r }. \]

Equivalently, its reciprocal-space coefficients are

\[ n_{ss'}(\boldsymbol Q) = \frac{1}{\Omega\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \sum_{\boldsymbol G} D_{\boldsymbol Gs, (\boldsymbol G-\boldsymbol Q)s'}(\boldsymbol k), \]

where both reciprocal vectors in each matrix element must belong to the retained orbital basis. In PAW, this contraction is the smooth auxiliary density; the physical density also contains projector-dependent one-center corrections. Those quantities are additional to \(\mathbf D(\boldsymbol k)\) and are defined in Projector-Augmented-Wave Method.

For a density represented without such augmentation, the scalar electron density and the three components of the spin-polarization density are obtained by writing \(\mathbf n(\boldsymbol r)=(n_{ss'}(\boldsymbol r))_{ss'}\) and taking

\[ n(\boldsymbol r) = \operatorname{tr}_s\mathbf n(\boldsymbol r), \qquad m^\alpha(\boldsymbol r) = \operatorname{tr}_s \left[ \sigma_\alpha\mathbf n(\boldsymbol r) \right]. \]

These are the same spin conventions as in Kohn–Sham Density Functional Theory.

Suppressing the weak \(\boldsymbol k\) dependence of \(N_{\mathrm{PW}}\), the full matrix \(\mathbf D(\boldsymbol k)\) contains \(O(N_{\mathrm{PW}}^2)\) spatial matrix elements, with additional explicit spin blocks, and is usually not formed explicitly. Implementations contract the occupied-state coefficients directly into the density and other required quantities. This computational choice does not change the definition of \(\mathbf D(\boldsymbol k)\).

Orbital and Density Cutoffs

If the orbital basis is bounded approximately by \(\lvert\boldsymbol k+\boldsymbol G\rvert\le G_{\mathrm{max}}\), products of orbitals contain Fourier components up to approximately \(2G_{\mathrm{max}}\). Expressed as a kinetic-energy scale, an exact density plane-wave expansion may therefore require a cutoff approaching

\[ E_{\mathrm{cut}}^{\mathrm{density}} \simeq 4E_{\mathrm{cut}}. \]

This relation describes the largest Fourier component generated by products; it does not prescribe a universal FFT grid. Practical grids must also prevent aliasing and represent the local potential, projector functions, and any augmentation densities. Pseudopotential and PAW datasets may consequently require density or augmentation grids substantially finer than the orbital grid.

Plane-wave convergence is systematic with respect to \(E_{\mathrm{cut}}\), but the required value is controlled by the hardest species and by the effective-core dataset. Brillouin-zone sampling, supercell size, and electrostatic treatment are independent convergence dimensions.

Spin and Symmetry

Each plane wave carries a two-component spin degree of freedom. A spinless nonmagnetic calculation may suppress this index and account for spin degeneracy through the occupations, but the general spinor formulation retains the full spinor coefficients

\[ \begin{pmatrix} c_{\boldsymbol G\uparrow,a}(\boldsymbol k)\\ c_{\boldsymbol G\downarrow,a}(\boldsymbol k) \end{pmatrix}. \]

Collinear magnetism makes the Hamiltonian block diagonal only when a common spin axis exists. Noncollinear fields and spin–orbit coupling mix these components. Spatial, magnetic, and antiunitary symmetries may relate coefficients or density matrices at different \(\boldsymbol k\) points, subject to the conditions stated in Periodic Systems. Reconstructing a density from an irreducible mesh requires the corresponding symmetry transformations, not only scalar multiplicity weights.

Scope and Numerical Character

The plane-wave basis does not depend on nuclear positions, so moving an atom does not generate atom-centered Pulay terms. At fixed energy cutoff, however, changing the cell changes the reciprocal lattice and the set of retained plane waves; incomplete cutoff convergence therefore produces Pulay stress. Derivatives of nonlocal projectors and PAW augmentation quantities remain part of forces and stress.

A molecule, slab, wire, or isolated defect treated with plane waves is still a periodic supercell calculation. Vacuum separation and, where needed, electrostatic corrections control interactions between periodic images.

References